Congressional Republicans are "demanding ransom" by refusing to fund the government unless Democrats agree to alter the health care law that passed in 2010, President Obama said today in a Rose Garden address.

"They're shutting down the government over an ideological crusade to deny affordable health care to many Americans," Obama said, flanked by Health and Human Services Secretary Kathleen Sebelius and beneficiaries of the health care law's new provisions.

"They are demanding ransom just for doing their job," he added. "This Republican shutdown did not have to happen."

The statement comes after Obama met in the Oval Office with 12 individuals who have either benefited from the Affordable Care Act's changes to health insurance coverage or plan to enroll in coverage through the new insurance marketplaces that came online today.

Obama lambasted Republicans for making the dismantling of the health care law the "centerpiece of their agenda," and he urged them to pass a budget and end the government shutdown.

"I will not give in to reckless demands by some in the Republican Party to deny affordable health insurance to millions," Obama said. "I will not negotiate over Congress' ability to pay bills it's already racked up.

"Pass a budget. End this shutdown."

With the next, potentially disastrous deadline right around the corner, the president urged lawmakers to raise the debt ceiling, warning "it'd be far more dangerous than a government shutdown, as bad as a shutdown is; it would be an economic shutdown."

On the first day of the health insurance marketplaces' opening to accept new enrollment, Obama acknowledged that the online system was experiencing "glitches" that he attributed to unexpected high demand.

"More than 1 million people visited healthcare.gov before 7 in the morning," Obama said. "We're going to be speeding things up in the next few hours to handle this demand exceeding anything we expected."

But on Capitol Hill, both Houses of Congress reinforced their original positions today despite a government shutdown that left nearly 800,000 federal employees out of work without pay this morning, and more than a million other "essential" employees on the job with many of their paychecks delayed.

With national parks and war memorials closed, House Republicans are now looking for ways to blunt the shutdown's damage, one politically sensitive part of the government at a time.

They plan to approve smaller spending bills that would fully fund veterans programs, the National Park Service and the District of Columbia.

Republicans held late-night votes in the House of Representatives in the early hours this morning to reaffirm their demands that the Democratic Senate agree to undermine President Obama's health care law in exchange for funding the government and that the Senate appoint representatives to a conference committee on the budget.

Democrats, in turn, gaveled the Senate into session this morning only to reject the House's bill, and Democratic leaders say they will only pass a bill to fund the government with no strings attached.

Meanwhile, with the stroke of midnight, the Affordable Care Act's health exchanges opened for business, allowing the uninsured to purchase health insurance.

And despite the Republican strategy to put the brakes on the health care bill, the law was implemented, starting today, regardless of the government shutdown.

The impact of the shutdown, however, is being felt in other ways. National parks in Washington and across the country are closed to the public and much of the government will be operating at reduced staffing levels that will lead to widespread delay.

People seeking Federal Housing Administration home loans will have to wait until the shutdown ends to secure mortgages. And in Washington, the courts have stopped issuing marriage licenses and performing weddings.

The political consequences could be grave for both parties in Congress, but especially for Republicans. According to an ABC News-Washington Post poll released Monday, 63 percent say they disapprove of Republicans' handling of the budget debate compared to 50 percent who disapprove of Obama's handling of the situation.

"The question is how fast does public pressure grow, particularly on Republicans," said Sarah Binder, expert in Congress and legislative politics and a professor at George Washington University. "There are federal employees everywhere."

"We underestimate some of the networks within which lawmakers will find themselves back home."